AMC 8 Daily Practice - New Patterns

Complete problem set with solutions and individual problem pages

Problem 1 Easy

Consider the following operation: a\oplus b=a-b^{2}. What is the value of  4 \oplus[5 \oplus(-2)]?

  • A.

    2

  • B.

    3

  • C.

    4

  • D.

    5

  • E.

    -2

Answer:B

According to the problem, a new operation is defined as a \oplus b = a - b^2.

To evaluate 4 \oplus [5 \oplus (-2)], we should firstly calculate the inner operation:    5 \oplus (-2) = 5 - (-2)^2= 5 - (4)= 1

Now substitute this result into the outer operation:    4 \oplus [5 \oplus (-2)] = 4 \oplus 1= 4 - (1)^2= 4 - 1= 3

Final result is \boxed{3}

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Problem 2 Easy

Define the operation A \otimes B = \frac{KA - B}{2}. Given that 5 \otimes 3 = 11, find the value of K.

  • A.

    3

  • B.

    4

  • C.

    5

  • D.

    8

  • E.

    11

Answer:C

Substitute A = 5 and B = 3 into the definition of \otimes:   5 \otimes 3 = \frac{K \cdot 5 - 3}{2} = 11

Multiply both sides by 2 to eliminate the denominator:   K \cdot 5 - 3 = 22

Add 3 to both sides:   K \cdot 5 = 25

Divide both sides by 5:   K = 5

Final Answer:  \boxed{5}

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Problem 3 Easy

Consider these two operations: 2 \# 4=2+2.5+3+3.5, 5 \# 3=5+5.5+6. What is the value of  10 \# 5?

  • A.

    10

  • B.

    12

  • C.

    55

  • D.

    50

  • E.

    15

Answer:C

The problem defines a custom operator \# such that:

2 \# 4 = 2 + 2.5 + 3 + 3.5   5 \# 3 = 5 + 5.5 + 6

By observing these examples:

The first number on the left side of = becomes the starting number of the sequence on the right side.

The second number on the left side of = indicates the number of terms in the sequence on the right side.

The sequence forms an arithmetic progression with a common difference of 0.5.

Therefore, for 10 \# 5=10 + 10.5 + 11 + 11.5 + 12 = 55$

The value of 10 \# 5 is \boxed{55}

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Problem 4 Easy

Let a and b be rational numbers. Define the operations:  a * b = \sqrt[3]{a} - \sqrt{b},  a \land b = a^{2} - b^{2}.

Compute (-1) \land (8 * 16).

  • A.

    -1

  • B.

    -2

  • C.

    -3

  • D.

    -4

  • E.

    -6

Answer:C

First, compute 8 * 16 using the first operation rule: 8 * 16 = \sqrt[3]{8} - \sqrt{16} = 2 - 4 = -2

Next, compute (-1) \land (-2) using the second operation rule: (-1) \land (-2) = (-1)^{2} - (-2)^{2} = 1 - 4 = -3

Final Answer:   \boxed{-3}

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Problem 5 Medium

An operation "* " is defined as follows: For any rational numbers x, y and z, x * x = 0, x * (y * z) = (x * y) + z, where "+" denotes ordinary addition. What is the value of 2023 * 2022?

  • A.

    0

  • B.

    1

  • C.

    2

  • D.

    3

  • E.

    4

Answer:B

 To compute 2023 * 2022:

1. Start with the definition 2022 * 2022 = 0.

2. Rewrite 2023 * 2022 by strategically adding and subtracting 2022.

2023 * 2022 = 2023 * 2022 + 2022 - 2022 =2023 * (2022 * 2022) -2022 = 2023 * 0 - 2022= 2023 * (2023 * 2023) - 2022 = (2023 * 2023) + 2023 - 2022 = 0+2023-2022 = 1

Final Answer:   \boxed{1}

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Problem 6 Easy

Define a new operation:   a \odot b = \begin{cases} a^{2} - b & \text{if } a \leq 0 \\ -a + b & \text{if } a > 0 \end{cases}.   For example, -2 \odot 4 = (-2)^{2} - 4 = 0, and 2 \odot 3 = -2 + 3 = 1. If x \odot 1 = -\frac{3}{4}, find the value of x.

  • A.

    -\frac{1}{2}

  • B.

    \frac{1}{2},-\frac{3}{4}

  • C.

    \frac{7}{4}

  • D.

    -\frac{1}{2},\frac{1}{2}

  • E.

    -\frac{1}{2},\frac{7}{4}

Answer:E

To find x, we consider the two cases based on the definition of \odot:

Case 1: x \leq 0

If x \leq 0, then x \odot 1 = x^{2} - 1 = -\frac{3}{4}  ， x = \pm \sqrt{\frac{1}{4}} = \pm \frac{1}{2}

Since x \leq 0 in this case, we discard x = \frac{1}{2} and retain x = -\frac{1}{2}.

Case 2: x > 0

If x > 0, then x \odot 1 = -x + 1 = -\frac{3}{4}, x = \frac{7}{4} > 0, this solution is valid.

Thus, the values of x are:   \boxed{-\frac{1}{2}} \quad \text{and} \quad \boxed{\frac{7}{4}}

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Problem 7 Easy

For two natural numbers a and b, their least common multiple (LCM) minus their greatest common divisor (GCD) is defined as a \pm b,  a \pm b = [a, b] - (a, b). What is the value of  125 \pm 25?

  • A.

    0

  • B.

    25

  • C.

    50

  • D.

    100

  • E.

    125

Answer:D

1. Compute the GCD of 125 and 25:      (125, 25) = 25

2. Compute the LCM of 125 and 25:      [125, 25] = 125

3. Subtract the GCD from the LCM:      125 \pm 25 = [125, 25] - (125, 25) = 125 - 25 = 100

Final Answer:  \boxed{100}

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Problem 8 Medium

For natural numbers a and n, the operation a \Delta n is defined as a^{n} + a^{n-1}.For example, 3 \Delta 2 = 3^{2} + 3 = 12. Compute the sum:   1 \Delta 2 + 2 \Delta 2 + 3 \Delta 2 + \cdots + 99 \Delta 2.

  • A.

    4950

  • B.

    9900

  • C.

    328530

  • D.

    333300

  • E.

    100000

Answer:D

The expression can be expanded as:   1 \Delta 2 + 2 \Delta 2 + 3 \Delta 2 + \cdots + 99 \Delta 2 = 1^{2} + 1 + 2^{2} + 2 + 3^{2} + 3 + \cdots + 99^{2} + 99 = 1^{2} + 2^{2} + 3^{2} + \cdots + 99^{2} + 1 + 2 + 3 + \cdots + 99

Using the formulas:   1^{2} + 2^{2} + 3^{2} + \cdots + n^{2} = \frac{n(n+1)(2n+1)}{6}， 1 + 2 + 3 + \cdots + n = \frac{n(n+1)}{2}

Substitute n = 99 into the formulas:   1^{2} + 2^{2} + 3^{2} + \cdots + 99^{2} + 1 + 2 + 3 + \cdots + 99 = \frac{99 \cdot 100 \cdot 199}{6} + \frac{100 \cdot 99}{2}=328350 + 4950 = 333300

The final result: \boxed{333300}

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Related Paper
AMC 8 Daily Practice - The Sum of a Finite Arithmetic Series AMC 8 Daily Practice - The Rule of Arithmetic Sequences AMC 8 Daily Practice - Consecutive Reduction AMC 8 Daily Practice - Calculation Tricks by Grouping
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